Curvature of Vector Bundles and Subharmonicity of Bergman Kernels

In a previous paper, [1], we have studied a property of subharmonic dependence on a parameter of Bergman kernels for a family of weighted L-spaces of holomorphic functions. Here we prove a result on the curvature of a vector bundle defined by this family of L-spaces itself, which has the earlier results on Bergman kernels as a corollary. Applying the same arguments to spaces of holomorphic sections to line bundles over a locally trivial fibration we also prove that if a holomorphic vector bundle, V , over a complex manifold is ample in the sense of Hartshorne, then V ⊗ det V has an Hermitian metric with curvature strictly positive in the sense of Nakano.